Notations.
Suppose that we are interested in evaluating an LLM \mathcal{M} on a task T using an evaluation metric E (such as “Accuracy”) under a format constraints C (such as “Bold the final answer.”) on n samples with the ground-truth answers \{y_{1},...,y_{n}\} and raw generated answers \{\hat{y}_{1},...,\hat{y}_{n}\} , where y_{i},\hat{y}_{i}\in\mathcal{Y} \forall i with \mathcal{Y} being the answer token sequence space. We denote F_{C} as the binary format-following evaluation function of C :

F_{C}(\hat{y}_{i})=\begin{cases}1,&if\ \hat{y}_{i}\ satisfies\ C.\ 0,&otherwise.\end{cases} (1)

From Eq. 1 , we define the Format Instruction-following (FI) Score, denoted as FI_{C} , as the percentage of generated outputs satsisfying C :

FI_{C}=\frac{\sum_{i=1}^{n}F_{C}(\hat{y}_{i})}{n}\cdot 100 (2)

Prior studies extensively focus on evaluating FI_{C} Zhou et al. ( 2023 ); Xia et al. ( 2024 ) . Our work further targets evaluating the performance of LLMs given the format constraints C . Under C , we denote Ext_{C}() as the rule-based answer extractor (or a mixture of extractors) to extract the final answer from \hat{y_{i}} for comparing it with y_{i} . We define:
two evaluation scores based on E :

SysE=\frac{1}{n}\sum_{i=1}^{n}(E(y_{i},Ext_{C}(\hat{y_{i}})).F_{C}(\hat{y_{i}})) (3)

Essentially, SysE quantifies the performance of \mathcal{M} on task T based on the generated answers that meet the format constraints C . For example, in Fig. 1 , Case (1) yields a SysE score of 0 , while Case (3) achieves 1 . This also shows that SysE may not accurately reflect the actual performance of \mathcal{M} on T , because Ext_{C}() may fail to extract the final answers from (correct) answers
dissatisfying C (e.g., Case (1) in Fig. 1 ).
We define the True Evaluation Score to address this. Assume that we have
an oracle extractor function OracExt_{C}() that can extract the
final answer from \hat{y_{i}} , regardless of whether \hat{y_{i}} fulfills C , we have:

TrueE=\frac{1}{n}\sum_{i=1}^{n}E(y_{i},OracExt_{C}(\hat{y_{i}})) (4)

TrueE measures the performance of \mathcal{M} on task T across all generated answers given the format constraints C , regardless of format satisfaction . In Fig. 1 , both Cases (1) and (3) achieve a true accuracy of 1 . This score is crucial for assessing the true performance of LLMs given the format.

Prior studies do not clearly differentiate between SysE and TrueE . In practice, measuring TrueE is challenging because OracExt_{C}() is unavailable. While researchers typically employ a mixture of methods to extract answers, this approach encounters two severe issues. First, these mixture-of-method extractors can be complex, unreliable, and often impractical for large-scale experiments with diverse formats like ours. Second, designing them to be reliable for complex formats such as medical reports can be impossible due to the countless potential errors. Another alternative is to assign a default value to Ext_{C}(\hat{y_{i}}) . While this can temporarily avoid cases \mathcal{M} fails to fulfill C , this is an incorrect practice since the default value may not be the actual output. Reliably measuring TrueE often requires human investigation (Lin et al., 2022 ) or the fine-tuning of evaluation models as scorers (Yang et al., 2024 ) , both of which are costly.

Nevertheless, TrueE is crucial for a fair evaluation of LLM performance bias across formats. Therefore, we propose a simple estimator of TrueE , denoted as EstTrueE :

EstTrueE=\begin{cases}SysE.\frac{100}{FI_{C}},&if\ FI_{C} eq 0.\ 0,&otherwise.\end{cases} (5)

When FI_{C}=0 , estimating EstTrueE becomes impossible. EstTrueE enables the fair format bias evaluation because normalizing SysE by FI_{C} prevents skewing comparisons of how different formats affect the LLM due to FI_{C} . It is especially useful for large-scale experiments since it is fully automatic. Let the EstTrueE margin of error be \epsilon with a confidence interval 1-\alpha and S_{C}=n\cdot FI_{C} as #generated answers satisfying C .

EstTrueE is consistent. Moreover, EstTrueE is reliable if and only if:

FI_{C}\geq\frac{1}{1+n\cdot\left(\frac{\epsilon}{v\cdot s}\right)^{2}} (6)

Moreover, we have:

\lim_{{FI_{C}}\to 100}{EstTrueE}={TrueE} (7)

where s^{2} is the sample variance of evaluation scores of generated answers satisfying C and v=t_{\alpha/2,S_{C}-1} is the critical value from the t-distribution with S_{C}-1 degrees of freedom.

In summary, we have proposed a consistent estimator EstTrueE of the true performance of LLMs measured by metric E under the format constraints (Def. 3.2 ). This estimator is essential because it: (1) ensures transparent and fair LLM performance evaluation across different formats; and (2) supports large-scale format bias evaluation. Note that a high score EstTrueE is only reliable iff FI_{C} is high enough ( Theorem 3.1 ). Henceforth, unless otherwise specified, EstTrueE is our primary metric for measuring model performance given format constraints. The proof of Theorem 3.1 is in Section B.1 .